Comprehensive notes for Chapter 3 Circular and Rotational Motion. Covers Angular Displacement, Velocity, Acceleration, Centripetal Force, Moment of Inertia, and Artificial Satellites.
Radian: The angle subtended at the center of a circle by an arc equal in length to the radius.
Relation: S = rθ (where θ is in radians).
1 Revolution = 2π rad = 360°.
1 rad ≈ 57.3°.
Angular Velocity (ω): Rate of change of angular displacement (ω = Δθ/Δt). Unit: rad/s.
Relation: v = rω.
Angular Acceleration (α): Rate of change of angular velocity (α = Δω/Δt). Unit: rad/s².
Relation: aₜ = rα.
The force required to bend the straight path of a particle into a circular path. It is always directed towards the center.
Formula: F_c = mv²/r = mrω².
Examples: Tension in a string, Gravitational force on satellites, Friction on banked roads.
The rotational analogue of mass. It measures the resistance of a body to change its rotational motion.
Formula: I = Σmr².
Torque and Inertia: τ = Iα.
Examples of I:
Hoop/Ring = mr²
Disc/Cylinder = ½ mr²
Sphere = 2/5 mr²
The cross product of position vector and linear momentum (L = r x p).
For a rigid body: L = Iω.
Law of Conservation: If no external torque acts on a system, the total angular momentum remains constant (L₁ = L₂ or I₁ω₁ = I₂ω₂).
Example: A diver curls their body (reduces I) to spin faster (increases ω).
Objects that orbit around the Earth. Gravity provides the centripetal force.
Orbital Velocity: v = √(GM/r).
Close to Earth (r ≈ R), v ≈ 7.9 km/s.
Time Period T = 2πr / v.
Created in spacecraft to simulate Earth's gravity and prevent weightlessness effects.
Achieved by rotating the spacecraft.
Frequency needed: f = (1/2π) √(g/R).